When Chiral Photons Meet Chiral Fermions: Photoinduced
Anomalous Hall Effects in Weyl Semimetals
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Chan, Ching-Kit, Patrick A. Lee, Kenneth S. Burch, Jung Hoon
Han, and Ying Ran. "When Chiral Photons Meet Chiral
Fermions: Photoinduced Anomalous Hall Effects in Weyl
Semimetals." Phys. Rev. Lett. 116, 026805 (January 2016). ©
2016 American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.116.026805
Publisher
American Physical Society
Version
Final published version
Accessed
Thu May 26 00:45:19 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/100969
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PRL 116, 026805 (2016)
week ending
15 JANUARY 2016
PHYSICAL REVIEW LETTERS
When Chiral Photons Meet Chiral Fermions:
Photoinduced Anomalous Hall Effects in Weyl Semimetals
1
Ching-Kit Chan,1 Patrick A. Lee,1 Kenneth S. Burch,2 Jung Hoon Han,3,* and Ying Ran2,†
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
3
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
(Received 27 September 2015; revised manuscript received 17 November 2015; published 15 January 2016)
The Weyl semimetal is characterized by three-dimensional linear band touching points called Weyl
nodes. These nodes come in pairs with opposite chiralities. We show that the coupling of circularly
polarized photons with these chiral electrons generates a Hall conductivity without any applied magnetic
field in the plane orthogonal to the light propagation. This phenomenon comes about because with all three
Pauli matrices exhausted to form the three-dimensional linear dispersion, the Weyl nodes cannot be gapped.
Rather, the net influence of chiral photons is to shift the positions of the Weyl nodes. Interestingly, the
momentum shift is tightly correlated with the chirality of the node to produce a net anomalous Hall signal.
Application of our proposal to the recently discovered TaAs family of Weyl semimetals leads to an orderof-magnitude estimate of the photoinduced Hall conductivity which is within the experimentally accessible
range.
DOI: 10.1103/PhysRevLett.116.026805
Introduction.—Previously thought to be an asset exclusive to massless elementary particles such as photons,
chirality has now become a defining emergent property of
electrons in such crystalline materials like graphene [1] and
the surface of topological insulators [2]. An exciting new
addition to the growing list of “chiral electronic materials”
is the Weyl semimetal [3–7]. In this three-dimensional
analogue of graphene, the Weyl nodes act as magnetic
monopoles in momentum space [8]. In sharp contrast to
the two Dirac nodes in graphene that behave more or
less as separate low-energy degrees of freedom, a pair of
Weyl nodes carrying opposite chiralities are inherently tied
together in a nonlocal manner through the “chiral anomaly”
mechanism which allows the dissipationless transfer
of charge between them [9]. The subject has undergone
vigorous research lately due to reports of material predictions, followed immediately by their synthesis and
confirmation of their Weyl characters by photoemission
experiments and nonlocal transport measurements. A
striking example of success along this line in recent years
is the transition metal monophosphide family that includes
TaAs, TaP, NbAs, and NbP [10–19].
In light of the rapid maturity of the Weyl semimetal
research, one should ask whether the unusual electronic
transport properties in both the static and dynamic regimes
of the Weyl semimetal can be exploited in high speed
electronics or to provide a new means of revealing dynamic
topological effects [18–20]. It was recently suggested that
exposing a two-dimensional Dirac material to a circularly
polarized (CP) light could lead to a novel form of Hall
effect due to the effective gap opening at the Dirac point
[21,22]. A natural question arises as to what happens when
0031-9007=16=116(2)=026805(5)
the three-dimensional, linearly dispersing bands of electrons couple to an intense CP light, as schematically
presented in Fig. 1, and whether a nontrivial Hall effect
can be induced. Here we present a direct consequence of
coupling CP light to the Weyl fermion: the appearance of
the anomalous Hall effect (AHE).
In this work, we show that even when the AHE is absent
due to symmetry reasons in typical Weyl materials, it must
be generally present in all Weyl systems when coupled
to a CP light source. Based on a low-energy effective WeylFloquet Hamiltonian analysis and symmetry considerations, we discuss the induced AHE as a generic and readily
FIG. 1. Schematic figure for a driven Weyl semimetal. Blue
ðIÞ
and red circles indicate Weyl nodes with opposite chiralities χ W .
The node positions are shifted by the chiral photons in a
chirality-dependent manner and the shift is proportional to the
square of the driving amplitude A. As a result, even though
the total momentum shift along the driving direction is zero
P
ðIÞ
( I Δkz ¼ 0), the overall Chern vector shift has a finite zP ðIÞ ðIÞ
component (δνz ¼ I χ W Δkz ≠ 0), resulting in a photoinduced
anomalous Hall conductivity σ xy [see Eq. (6)].
026805-1
© 2016 American Physical Society
PRL 116, 026805 (2016)
observable phenomenon in Weyl semimetals. We further
demonstrate this effect using a concrete microscopic model,
and apply our study to the TaAs family of Weyl semimetals
with an estimation of the experimental feasibility.
Other interesting works have addressed the driven threedimensional Dirac electrons [23,24] or gapped systems
[25,26]. However, neither systems enjoy the unique topological protection of the Weyl nodes, whose response,
under a CP beam, results in the experimental signature of
the AHE as we explain below.
General tight-binding model analysis.—The low-energy
Hamiltonian for a single Weyl node can be parameterized most generally by a real 3-vector α~ and a 3 × 3
real matrix β:
H W ð~qÞ ¼ qi αi σ 0 þ qi βij σ j ;
ð1Þ
with a canonical 3D linear dispersion EW ð~qÞ ¼ αi qi P P
½ j ð i qi βij Þ2 1=2 . Einstein summation is assumed in
the momentum qi and the Pauli matrix σ j . The chirality
of the Weyl node is given by the sign of the determinant
of β: χ W ¼ sgn½DetðβÞ. For a practical band structure,
this 2 × 2 Weyl Hamiltonian is embedded within a large
N-band Hamiltonian as a low-energy sector. In order to
have the most general consideration of the photoinduced
AHE, we therefore consider an N-band tight-binding
~ The expansion of HðkÞ
~ around a Weyl
Hamiltonian HðkÞ.
node at k~W to first order in q~ ¼ k~ − k~W can be put in the
block form:
U† Hðk~W þ q~ ÞU ¼ Hlin ð~qÞ þ Oðq2 Þ;
HW ð~qÞ
qi Ci
;
Hlin ð~qÞ ¼
qi C†i
D0 þ qi Di
ð2Þ
where the matrix D0 gives the (N − 2) high-energy states at
k~ ¼ k~W . Eigenvectors at k~ ¼ k~W are used to construct the
unitary matrix U whose first two columns correspond to the
two zero-energy states. In general, the 2 × ðN − 2Þ matrices
Ci mix the Weyl bands with high-energy bands linearly in q~
and should not be ignored.
Without loss of generality, we consider an incident
electromagnetic wave in the z direction with the vector
~ ¼ ðAx cosðωtÞ; Ay sinðωt þ ϕÞ; 0Þ. Following
potential AðtÞ
standard routes [21], the Peierls substitution Hlin ð~qÞ →
~
Hlin ð~q þ eAðtÞÞ
and averaging over one drive cycle leads
to the effective Hamiltonian [27]:
Heff ð~qÞ ¼
HWF ð~qÞ
week ending
15 JANUARY 2016
PHYSICAL REVIEW LETTERS
OðqÞ þ OðA2 Þ
OðqÞ þ OðA2 Þ D0 þ OðqÞ þ OðA2 Þ
; ð3Þ
e2 Ax Ay cos ϕ
HWF ð~qÞ ¼ HW ð~qÞ −
ω
i
†
†
× ϵijk βxi βyj σ k − ðCx Cy − Cy Cx Þ : ð4Þ
2
Higher-order terms Oðq2 Þ þ OðqA2 Þ þ OðA4 Þ are ignored.
Clearly, the photoinduced effect is maximized for a CP
light when cos ϕ ¼ 1, and vanishes for a linearly polarized beam.
The CP light has two contributions to the Weyl-Floquet
Hamiltonian in Eq. (4). The first (β2 ) term originates
entirely from the two-band Weyl Hamiltonian HW ð~qÞ,
while the second term proportional to Cx C†y − Cy C†x is
due to high-energy band mixing and must be kept in
the construction of a general Weyl-Floquet effective model.
For later convenience, we reparameterize −ði=2ÞðCx C†y −
Cy C†x Þ ¼ κ 0 σ 0 þ κi σ i with four real numbers κ i ¼
ð−i=4ÞTr½σ i ðCx C†y − Cy C†x Þ. Therefore, the influence of
the drive to A2 order on the 2 × 2 Weyl band can be
summarized as
HWF ð~qÞ ¼ ðqi − δqi Þαi σ 0 þ ðqi − δqi Þβij σ j − δμ · σ 0 ;
e2 Ax Ay cos ϕ
½ϵjkl βxj βyk þ κ l ðβ−1 Þli ;
ω
e2 Ax Ay cos ϕ
δμ ¼ −δ~q · α~ þ
κ0 :
ω
δqi ¼
ð5Þ
δqi and δμ represent the shifts of Weyl momenta and
chemical potential, respectively.
The photoinduced Weyl node shift has an immediate
consequence on the electronic transport. According to
Ref. [6], for a Weyl system where the chemical potential
situates at the nodes (i.e., μ ¼ 0), the leading order change
of the anomalous Hall conductivity is governed by the
momentum shifts of every Weyl node as
δσ ij ¼
e2
ϵ δν ;
2πh ijk k
where δνk ¼
X
I
ðIÞ
ðIÞ
χ W · δqk :
ð6Þ
Different Weyl nodes are labeled by the superscript I and δ~ν
is the change of the Chern vector [6]. Provided that the
ðIÞ
momentum shifts δqk among Weyl nodes of different
ðIÞ
chiralities χ W do not cancel out, one would expect the AHE
induced at the A2 order to be proportional to the intensity of
the incident beam. Note that the photoinduced self-doping
(δμ ≠ 0) of the Weyl nodes contributes to δσ ij at a higher
order Oðδμ2 Þ ¼ OðA4 Þ [6] of the drive intensity, which is
negligible. If κ~ðIÞ ¼ 0, one can show the z-component of
the Chern vector shift to be
δνz ¼
where the 2 × 2 Weyl-Floquet part reads
026805-2
e2 Ax Ay cos ϕ X ½CofðβðIÞ Þzi 2
;
ω
j detðβðIÞ Þj
I;i
ð7Þ
PHYSICAL REVIEW LETTERS
PRL 116, 026805 (2016)
where CofðβðIÞ Þij is the ij element of the cofactor matrix of
βðIÞ . The alternating signs of χ ðIÞ between a node and an
antinode, or a monopole and an antimonopole, are precisely
ðIÞ
cancelled by the same sign change in δqz . As such, each
Weyl node has a positive-definite contribution to δνz .
Consequently, a finite Hall conductivity is induced in the
plane perpendicular to the incident beam and its sign is
determined by the photon chirality through σ xy ∝ cos ϕ.
In practice, the pre-drive chemical potential μ may not
cross the Weyl node as we have discussed so far. As long
as μ before the drive lies in the linearly dispersive regime
near the node, one can show that the our resultant δσ ij is
unaffected [27]. In fact, using a frequency of excitation
0 < ω < 2μ for slightly electron-doped Weyl node would
ensure no absorption, as assumed here, and thus should
lead to a more straightforward analysis of the experimental
results regarding the photoinduced AHE.
Symmetry consideration for multiple Weyl nodes.—In a
typical Weyl material, there are multiple sets of Weyl nodes
related by the space group and time-reversal (TR) symmetries [5,10]. The parameters α~ , β, and Ci characterizing
symmetry-related nodes transform accordingly. We now
derive their transformation rules and discuss how the nodal
shifts δ~q and δμ among the symmetry-related nodes are
correlated.
ðIÞ
A pair of Weyl points k~W (with I ¼ 1, 2) related by some
ð1Þ
ð2Þ
space-group symmetry satisfy Rk~W ¼ k~W , where R ∈ Oð3Þ
defines the particular symmetry operation. The parameters
quantifying the Weyl nodes transform according to
ð2Þ
ð1Þ
αi ¼ Rij αj ;
ð2Þ
ð1Þ
βij ¼ Rik βkj ;
ð2Þ
ð1Þ
Ci ¼ Rij Cj :
ð2Þ
ð1Þ
ð2Þ
ð1Þ
βij ¼ Rik βkl ξlj ;
ð2Þ
ð1Þ
Because of the presence of ξ ¼ ðξlj Þ ¼ diagð1; −1; 1Þ, the
chiralities of these two nodes are related by −Det½R. For
instance, the TR operation with R ¼ diagð−1; −1;−1Þ gives
the same chirality for the nodes at k~W and Rk~W ¼ −k~W .
Some proposed Weyl semimetals break TR symmetry
but preserve inversion (I). Our symmetry analysis predicts
ð2Þ
ð1Þ
α~ ð2Þ ¼ −~αð1Þ , βð2Þ ¼ −βð1Þ , κ 0 ¼ κ0 and κ~ð2Þ ¼ κ~ð1Þ for a
pair of I-related nodes and thus, using Eq. (5),
δ~q
ð1Þ
¼ −δ~q ;
ð2Þ
δμ
of them. Hence, from Eq. (6), these two nodes contribute
additively to the Hall conductivity. A similar conclusion
holds for I symmetry breaking Weyl materials for which
TR is a good symmetry. In this case we find α~ ð2Þ ¼ −~αð1Þ ,
ð2Þ
ð1Þ
βð2Þ ¼ −βð1Þ · ξ, κ0 ¼ −κ 0 and κ~ð2Þ ¼ −ξ · κ~ð1Þ for TRrelated nodes, and thus
Ci ¼ Rij ½Cj :
ð9Þ
ð2Þ
FIG. 2. (a) The 12 Weyl nodes in the first Brillouin zone of the
undriven lattice Weyl model studied in Ref. [30]. (b), (c), and
(d) Photoinduced Weyl node shifts in the presence of a CP drive
along the z direction. Nodes with positive (negative) chirality are
represented by red (blue) dots, and (I) labels the I-th Weyl node
[see Eq. (12)]. Open circles denote the undriven Weyl node
positions. Only projections on the kz ¼ 2π, ky ¼ 2π, and kx ¼ 2π
planes are shown as we increase A2 from 0 to 0.3. The shifts in (b)
are magnified 10 times for a better visualization. Details of the
model calculation can be found in Ref. [27]. Parameters used:
ðt; ϵ; λ; ωÞ ¼ ð1; 3; 1; 1.1Þ.
ð8Þ
As a result, the chiralities of nodes related by a rotation
(Det½R ¼ þ1) are the same, whereas a mirror or inversion
operation (Det½R ¼ −1) produces a sign difference. On
the other hand, nodes related by an anti-unitary symmetry
involving TR operation satisfy
αi ¼ Rij αj ;
week ending
15 JANUARY 2016
ð1Þ
¼ δμ ðI-relatedÞ: ð10Þ
Since these two nodes must have opposite chiralities
(Det½R ¼ −1), the product χ ðIÞ δ~qðIÞ is identical for both
δ~qð2Þ ¼ δ~qð1Þ ;
δμð2Þ ¼ −δμð1Þ ðTR -relatedÞ:
ð11Þ
Since χ ð1Þ ¼ χ ð2Þ , Eq. (6) once again predicts constructive
contributions from the TR-related nodes.
Lattice model analysis.—To illustrate our findings, we
now proceed to study a concrete lattice model for a Weyl
semimetal of the inversion symmetry breaking type [30]. The
model is parameterized by the hopping (t), spin-orbit coupling
(λ), and inversion breaking potential (ϵ) [27]. The bare system
possesses 12 Weyl nodes labeled as [see Fig. 2(a)]:
ð1;2Þ
k~W− ¼ ðk0 ; 0; 2πÞ;
ð3;4Þ
k~Wþ ¼ ð0; k0 ; 2πÞ;
ð5;6Þ
k~Wþ ¼ ðk0 ; 2π; 0Þ;
ð7;8Þ
k~W− ¼ ð2π; k0 ; 0Þ;
ð9;10Þ
k~W− ¼ ð0; 2π; k0 Þ;
026805-3
ð11;12Þ
k~Wþ ¼ ð2π; 0; k0 Þ;
ð12Þ
PHYSICAL REVIEW LETTERS
PRL 116, 026805 (2016)
week ending
15 JANUARY 2016
along with their chiralities as subscripts and a characteristic
momentum k0 ¼ 2 sin−1 ½ϵ=ð4λÞ. In the presence of a CP
drive, each node undergoes a momentum shift given by
ð3;4Þ
¼ δ~qþ
−δ~qð1;2Þ
−
ð5;6Þ
δ~qþ
¼ −δ~qð7;8Þ
−
ð11;12Þ
−δ~qð9;10Þ
¼ δ~qþ
−
e2 A2 8ηð1 − ηÞλ2
ẑ;
ω
ϵ
e2 A2 8ηð1 þ ηÞλ2
¼
ẑ;
ω
ϵ
e2 A2 ϵt2
¼
ẑ;
ω 16ηλ2
¼
ð13Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
with η ¼ 1 − ϵ2 =16λ2 . The momentum shifts clearly correlate with the chiralities. The photoinduced self-dopings are
−δμð9Þ ¼ δμð10Þ ¼ δμð11Þ ¼ −δμð12Þ ¼
e2 A2 ϵ2
;
ω 16
ð14Þ
and δμð1−8Þ ¼ 0. The anomalous Hall conductivity obtained
is (up to A2 order):
e2 e2 A2 64ηλ2
ϵt2
δσ xy ¼
;
ð15Þ
þ
2πh ω
ϵ
4ηλ2
which agrees with the expectation of the low-energy analysis.
So far all our discussion have been done within the
effective Hamiltonian scheme valid for a small driving
amplitude. A more stringent check free from this
assumption is offered by solving the full Floquet
Hamiltonian whose matrix elements are hn0 jH F jni ¼
Hn−n0 þ nωδn;n0 I 4 with nðn0 Þ denoting the nðn0 Þ-th
Floquet band. The topological character of the Weyl nodes
indeed protects them against the drive, and only their
locations are shifted continuously in a chirality-dependent
way. Figure 2(b)–(d) demonstrate the Weyl node shifts
obtained from diagonalizing the full Floquet Hamiltonian
as we increase the driving field strength. It is obvious that
nodes with opposite chirality have opposite shifts in kz ,
leading to an imbalance of δνz and eventually, a net
anomalous Hall conductivity. The overall field dependence
of the sum of Δkz weighted by chiralities is shown in
Fig. 3(a), which exhibits the anticipated A2 increment (we
have put e ¼ 1). This quantity equals to the Chern vector
δνz in the small A regime [Eq. (6)]. On the other hand,
the resultant self-doping shows the expected A2 dependence as illustrated in Fig. 3(b). However, it only corresponds to a higher order effect and does not affect the
photoinduced AHE.
Application to TaAs family of Weyl semimetals.—TaAs
was recently shown to host 24 Weyl nodes in the Brillouin
zone [10,11,15]. There are two nonequivalent sets of Weyl
points [10] that we denote as the P set consisting of 8 nodes
at (kPx , kPy , 0) and (kPy , kPx , 0), and the Q set with
Q
Q
Q
Q
Q
16 nodes at (kQ
x , ky , kz ) and (ky , kx , kz ). The
Q
Q
values of kPx , kPy , kQ
x , ky , kz for various TaAs family of
materials can be found in Ref. [10]. Applying the symmetry
FIG. 3. The field dependence for (a) the total Δkz weighted by
chiralities and (b) the absolute self-doping jδμj averaged over 12
Weyl nodes for the driven Weyl lattice model. Same parameters
used as in Fig. 2.
transformation rules derived before, we find a simple rule
for the node shifts for both Weyl sets:
ðP;IÞ
¼ χ ðP;IÞ δqPz ;
1 ≤ I ≤ 8;
ðQ;IÞ
¼ χ ðQ;IÞ δqQ
z ;
1 ≤ I ≤ 16;
δqz
δqz
ð16Þ
PðQÞ
∝ A2 is the same for all 8(16) symmetrywhere δqz
related nodes. The z-direction momentum shift faithfully
follows the chirality χ ðIÞ of each Weyl node, whereas the
same is not true for the shifts in the xy plane. Therefore, the
overall photoinduced AHE for the TaAs family is
σ xy ¼
e2
ð8 × δqP þ 16 × δqQ Þ:
2πh
ð17Þ
Experimental realization.—To guide future experimental
efforts to observe these effects, we discuss the conditions
needed and likelihood of success. In the Supplemental
Material we detail our calculations of the magnitude of the
Hall voltage [27], where we assume the use of a CW CO2
laser, with a roughly constant illumination across the
contacts, spaced ≈100 μm apart. Using established values
for the Fermi velocity [13,16,20], reflectance, optical [31]
and dc conductivities [18], we find a 100 nm thick film [32]
of TaAs would produce a Hall signal ≈130 nV at room
temperature, with a dc current of 1 A and 1 W laser power.
We note that the small penetration depth requires the use of
thinner samples to ensure most of the current is modulated
by the light (the induced voltage is inversely proportional to
the square of the thickness). While such signals should be
straight forward to detect, even higher values may be
possible through the use of pulsed lasers with higher peak
electric fields, with the Hall signal detected through
Faraday rotation measurements.
Conclusion.—An assortment of analyses based on
Floquet theory is carried out on models of Weyl semimetals
to argue that the AHE is induced generically by applying an
ac electromagnetic field of a definite chirality, in the plane
orthogonal to the incident beam. This photoinduced AHE
originates from the remarkable monopole nature of Weyl
nodes. The induced Hall conductivity scales linearly
026805-4
PRL 116, 026805 (2016)
PHYSICAL REVIEW LETTERS
and continuously with the field intensity. The nodal shift
itself may be observable by the pump-probe ARPES, while
the photoinduced AHE can be detected via dc transport
or Faraday rotation experiment on films. Conceivably,
the ac-field-driven AHE may exist in other materials
possessing non-trivial band topology such as ferromagnetic
metals [33].
P. A. L. acknowledges support from DOE Grant No. DEFG02-03- ER46076. K. S. B. acknowledges support from
the National Science Foundation under Grant No. DMR1410846. J. H. H. is supported by the NRF grant
(No. 2013R1A2A1A01006430). Y. R. is supported by
the Alfred P. Sloan fellowship and National Science
Foundation under Grant No. DMR-1151440.
[15]
[16]
[17]
[18]
*
hanjh@skku.edu
ying.ran@bc.edu
A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.
Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
(2010).
C. Herring, Phys. Rev. 52, 365 (1937).
S. Murakami, New J. Phys. 9, 356 (2007).
X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov,
Phys. Rev. B 83, 205101 (2011).
K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84, 075129
(2011).
A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205
(2011).
Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R.
Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y.
Tokura, and K. Terakura, Science 302, 92 (2003).
H. Nielsen and M. Ninomiya, Phys. Lett. 130B, 389
(1983).
H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai,
Phys. Rev. X 5, 011029 (2015).
S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang,
B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S.
Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Commun. 6,
7373 (2015).
S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,
C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M.
Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil,
F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan,
Science 349, 613 (2015).
B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J.
Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z.
Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013
(2015).
C.-C. Lee, S.-Y. Xu, S.-M. Huang, D. S. Sanchez, I.
Belopolski, G. Chang, G. Bian, N. Alidoust, H. Zheng,
†
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
026805-5
week ending
15 JANUARY 2016
M. Neupane, B. Wang, A. Bansil, M. Z. Hasan, and H. Lin,
Phys. Rev. B 92, 235104 (2015).
B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C.
Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N.
Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H.
Ding, Nat. Phys. 11, 724 (2015).
L. X. Yang, Z. K. Liu, Y. Sun, H. Peng, H. F. Yang, T.
Zhang, B. Zhou, Y. Zhang, Y. F. Guo, M. Rahn, D.
Prabhakaran, Z. Hussain, S.-K. Mo, C. Felser, B. Yan,
and Y. L. Chen, Nat. Phys. 11, 728 (2015).
S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R.
Chang, H. Zheng, V. N. Strocov, D. S. Sanchez, G. Chang,
C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M.
Huang, B. Wang, A. Bansil, H.-T. Jeng, T. Neupert, A.
Kaminski, H. Lin, S. Jia, and M. Zahid Hasan, Nat. Phys.
11, 748 (2015).
X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H.
Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen,
Phys. Rev. X 5, 031023 (2015).
C. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong,
N. Alidoust, C.-C. Lee, S.-M. Huang, H. Lin, M. Neupane,
D. S. Sanchez, H. Zheng, G. Bian, J. Wang, C. Zhang, T.
Neupert, M. Z. Hasan, and S. Jia, arXiv:1503.02630.
C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas,
I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, Z. Liu,
Y. Chen, W. Schnelle, H. Borrmann, Y. Grin, C. Felser, and
B. Yan, Nat. Phys. 11, 645 (2015).
T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik,
Science 342, 453 (2013).
A. Narayan, Phys. Rev. B 91, 205445 (2015).
S. Ebihara, K. Fukushima, and T. Oka, arXiv:1509.03673.
N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490
(2011).
R. Wang, B. Wang, R. Shen, L. Sheng, and D. Y. Xing,
Europhys. Lett. 105, 17004 (2014).
See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.116.026805 for more
details about the effective Weyl-Floquet Hamiltonian, the
lattice model calculation, realistic experimental estimations,
and the doping effect, which includes Refs. [28,29].
J. H. Shirley, Phys. Rev. 138, B979 (1965).
P. Delplace, J. Li, and D. Carpentier, Europhys. Lett. 97,
67004 (2012).
T. Ojanen, Phys. Rev. B 87, 245112 (2013).
B. Xu, Y. M. Dai, L. X. Zhao, K. Wang, R. Yang, W.
Zhang, J. Y. Liu, H. Xiao, G. F. Chen, A. J. Taylor, D. A.
Yarotski, R. P. Prasankumar, and X. G. Qiu, arXiv:1510.00470.
The proposed film has a thickness larger than the lattice
spacing by 2 orders of magnitude so that the momentum
along the width is still a good quantum number.
N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N.
P. Ong, Rev. Mod. Phys. 82, 1539 (2010).